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  • Springer  (113,630)
  • 1990-1994
  • 1980-1984  (43,712)
  • 1970-1974  (69,918)
  • 1930-1934
  • 1981  (43,712)
  • 1972  (36,116)
  • 1971  (33,802)
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  • 1990-1994
  • 1980-1984  (43,712)
  • 1970-1974  (69,918)
  • 1930-1934
Year
  • 1
    Electronic Resource
    Electronic Resource
    Contributions to the theory of organismic sets, II: Aggressiveness and submissiveness (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 49-54 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the theory of organismic sets (Bull. Math. Biophysics,31, 159–198, 1969) we considered organisms as sets endowed with certain “activities,” the latter’s resulting in a set of “products.” Those products may be of a material nature, like a hormone secreted by a cell, or of a non-material nature, like a feeling or an attitude. In the present paper aggressiveness and submissiveness are considered as such non-material products of the activities of the brain cells. A general description of aggressiveness and submissiveness is given in terms of organismic sets. Cycles in “peck order” are thus naturally explained.
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    URL: http://dx.doi.org/10.1007/BF02476664
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  • 2
    Electronic Resource
    Electronic Resource
    Contribution to the theory of organismic sets, III (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 55-66 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern〉1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.
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    URL: http://dx.doi.org/10.1007/BF02476665
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  • 3
    Electronic Resource
    Electronic Resource
    “A stochastic solution of the 3-body problem in chemical kinetics. The termolecular stochastic process I” (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 67-81 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It appears to be axiomatic that termolecular and higher order reactions occur relatively rarely. The basis for this judgment seems to lie in the supposition that successful 3-Body collisions of 3 interactive species of molecules cannot occur frequently enought to account for chemical or biochemical transformation. In order to provide a more complete mathematical framework than now exists for examining this hypothesis the probability of effective termolecular “δ-collisions” as a function of time is derived. This amounts to adding to the class of reactions for which stochastic models are now available the termolecular reaction. In common with the unimolecular and bimolecular cases this process is seen to satisfy the criterion of consistency-in-the-mean with respect to deterministic formulations. It is planned next to use the termolecular process and the lower order processes in computer-assistedin numero experimental studies aimed at comparing alternative mechanisms of reaction.
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    URL: http://dx.doi.org/10.1007/BF02476666
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  • 4
    Electronic Resource
    Electronic Resource
    Maximum likelihood estimation for stochastic first order reactions (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 83-96 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Small sample properties of the maximum likelihood estimator for the rate constant of a stochastic first order reaction are investigated. The approximate bias and variance of the maximum likelihood estimator are derived and tabulated. If observations of the system are made at timesiτ,i=1, 2, ...,N; τ〉0, the observational spacing τ which minimizes the approximate variance of the maximum likelihood estimator is found. The non-applicability of large sample theory to confidence interval derivation is demonstrated by examination of the relative likelihood. Bartlett’s method is employed to derive approximate confidence limits, and is illustrated by using simulated kinetic runs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476667
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  • 5
    Electronic Resource
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    Organismic supercategories and qualitative dynamics of systems (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 339-354 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476778
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  • 6
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    Some realizations of (M, R)-systems and their interpretation (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 303-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some years ago (Rosen 1958a, b; 1959) we described a class of metaphorical, relational paradigms for cellular activity which we termed (M, R)-systems. A sizable amount of subsequent work, to be itemized below, has been devoted to an exploration of some of the properties of these systems. The main purpose of the present paper is to put this class of paradigms into a general system-theoretic perspective, with a particular view to appraising the relation between the type of system description embodied in the (M, R)-system and other kinds of physical and mathematical descriptions of cellular systems. Thus, the principal aim is to establish the relationships and connections between the global relational formalism embodied in the (M, R)-systems and the empirical descriptions which still represent the bulk of our biological knowledge.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476776
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  • 7
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    Relational forces and structures produced by them (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 321-338 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract After giving a brief review of the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967;31, 159–198, 1969), in which the concept of relational forces, introduced earlier (Bull. Math. Biophysics,28, 283–308, 1966a) plays a fundamental role, the author discusses examples of possible different structures produced by relational forces. For biological organisms the different structures found theoretically are in general agreement with observation. For societies, which are also organismic sets as discussed in the above references, the structures can be described only in an abstract space, the nature of which is discussed. Different isomorphisms between anatomical structures, as described in ordinary Euclidean space, and the sociological structures described in an abstract space are noted, as should be expected from the theory of organismic sets.
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    URL: http://dx.doi.org/10.1007/BF02476777
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  • 8
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    Probabilistic and multiple regression modeling of the biologist’s decision processes and its implications for medical diagnosis (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 439-449 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Current psychological research into the inference (diagnostic) process is briefly reviewed, using as a vehicle an investigation of the prediction of the probability of success of hypothetical applicants to a graduate program in biology. Brunswik’s lens model and multiple regression analysis are used, as is a Bayesian approach. Four judges’ (biologists’) predictions are analyzed. Some general conclusions about inference, drawn from the current data in psychology, are presented.
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    URL: http://dx.doi.org/10.1007/BF02476785
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  • 9
    Electronic Resource
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    Anin numero study of glucose-insulin interaction (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 451-462 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model has been developed to simulate the glucose-insulin interaction following a glucose load such as occurs in an IVGTT. This model differs from earlier models in that the insulin response to glucose loading is a recurring all or none threshold response. The model has been simulated on a digital computer using the digital analog simulation language CSMP.
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    URL: http://dx.doi.org/10.1007/BF02476786
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  • 10
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    Wave propagation in bone media (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 463-479 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.
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    URL: http://dx.doi.org/10.1007/BF02476787
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  • 11
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    Fourth International Biophysics Congress (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 481-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476788
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  • 12
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    Obituary of Nicolas Rashevsky (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. i 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02477019
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  • 13
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    On the stochastic theory of compartments: I. A single-compartment system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 53-63 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.
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    URL: http://dx.doi.org/10.1007/BF02477024
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  • 14
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    A remark on sensory deprivation from the point of view of organismic sets (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 439-441 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that from the definition of organismic sets (Rashevsky,Organismic Sets. Some Reflections on the Nature of Life and Society, Holland, Michigan, Mathematical Biology, Inc. and Grosse Pointe, Michigan, J. M. Richards Laboratory) a complete sensory deprivation of an organismic set of ordern=2 should result in malfunctioning of the set. A generalization to higher order sets is suggested.
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    URL: http://dx.doi.org/10.1007/BF02476706
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  • 15
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    Optimal branching structure of the vascular tree (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 431-438 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Optimality in branching structure of the vascular tree was studied. Analysis on its physiological roles as the duct system for blood supply to the capillaries predicted that the vascular tree should be constructed with minimum volume under restriction of determinant pressure, flow and location at the origin and the terminals. Mathematical derivations of this conditional extremum problem yielded some equations expressing the relations between the radii of the branches and their branching angles, which provided numerical solutions for branching points of bi- and poli-terminal minimum volume trees. Comparison of the peritoneal vascular tree in a dog with the minimum volume one computed under the same restrictive conditions showed good agreement in their branching structure.
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    URL: http://dx.doi.org/10.1007/BF02476705
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  • 16
    Electronic Resource
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    First order kinetics in a distributed model (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 443-456 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In order to determine the kinetics of passage of a substance through an organ containing a tangle of vessels, we study the response of a tube to various inflows (perfusion, brief injection, ...). The introduction of the catabolic terms and of the spatial dependence between bulk concentration and surface concentration allows one to account for the difference of arteriovenous concentrations observed experimentally for many metabolites. The relationships between the physico-chemical parameters of the organ and the operational parameters of the model demonstrate the importance of the transit time through the considered vessels. If one considers the different pathways as independent, the introduction of the transit time distribution for an inert substance enables one to compute the response of the organ analytically or by recurrence, using convolution. The parameters of the model can be obtained by the moments method.
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    URL: http://dx.doi.org/10.1007/BF02476707
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  • 17
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    Stochastic simulation of the gas diffusion in the air phase of the human lung (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 457-466 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.
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    URL: http://dx.doi.org/10.1007/BF02476708
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  • 18
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    A new mathematical model of the respiratory center (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 467-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A new mathematical model of the oscillatory behavior of the respiratory center has been developed based upon published records of neuronal activity during respiration in the pons and medulla. In contrast with a previous model, four, rather than two, networks are assumed to interact in the respiratory center so as to produce the respiratory oscillation. A mathematical description of this interaction, in the form of a set of four first-order, nonlinear, coupled differential equations, is derived; the behavior of the solutions of this system is studied qualitatively, and expressions for the durations of the inspiratory and expiratory phases are obtained in terms of some parameters. It is found that central and chemical influences drive the medullar neurons to a position somewhere between saturation and full cutoff, and the pontine neurons deeply into cutoff. The control of the duration of the different phases by these chemical and central means is discussed. In order to effect a decrease in the magnitude of the various times, the neurons have to be driven towards operating points of higher central facilitation.
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    URL: http://dx.doi.org/10.1007/BF02476709
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  • 19
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    A mathematical model for calcium homeostasis (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 483-502 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.
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    URL: http://dx.doi.org/10.1007/BF02476710
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  • 20
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    The struggle for life: III. A predator-prey chain (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 521-532 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.
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    URL: http://dx.doi.org/10.1007/BF02476712
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  • 21
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    Mathematical models for exchange of substances in regional vascular beds— Measurement of the rates within vivo counters (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 503-520 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.
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    URL: http://dx.doi.org/10.1007/BF02476711
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  • 22
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    A numerical experiment on the determination of kinetic rate constants in the presence of diffusion (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 547-558 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two chemicals,A andB, are allowed to diffuse together and a reaction described by $$A + B\mathop \rightleftharpoons \limits_{K_{ - 1} }^{K_1 } C$$ is allowed to proceed. This system is described mathematically by a system of partial differential equations. A numerical procedure is presented to find the rate constants ofK 1 andK −1. A systematic analysis of the effects of errors is also presented.
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  • 23
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    Observations on cardiac energetics: I. Theoretical introduction (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 533-546 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Equations are developed to describe the energy expenditure of the human heart. As well as the external potential and kinetic energy terms, general consideration is given to other possible avenues of energy consumption. Emphasis is placed upon using mathematical variables which are readily available for experimental verification. The errors involved in assuming that mean values for the physiological parameters give reasonable estimations for the external mechanical performance are examined, and a theoretical estimation for the discrepancy in the kinetic component is presented. Logical extension of the mathematical derivation leads to a determination of cardiac external mechanical efficiency and clearly demonstrates the significance of the ventricular pressure-volume loop in this context. Finally, experimental procedures are suggested to clarify further some of the conclusions reached through the theoretical analysis.
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    URL: http://dx.doi.org/10.1007/BF02476713
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  • 24
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    Organismic sets. Sex in animal and human societies (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 559-563 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The question is discussed as to the reason why some animal societies, such as bees or ants, are sexually differentiated, that is, onlysome of its members are exhibiting reproducing activities. It is indicated that human society may be on its way to such a sexual differentiation which may eventually come.
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  • 25
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    Some biochemical threshold mechanisms (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 565-565 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    Announcement (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 567-567 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 27
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    Multibreath tracer species dynamics in the lung (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 1-19 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function. Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood. The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood. Models currently in the literature are shown to be special cases of the model presented here.
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    URL: http://dx.doi.org/10.1007/BF02460935
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  • 28
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    Global stability of population models (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 47-58 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.
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    URL: http://dx.doi.org/10.1007/BF02460938
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  • 29
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    The logistic equation with a diffusionally coupled delay (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 125-140 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.
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    Uncertainty in physiology and physics (1981)
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    Bulletin of mathematical biology 43 (1981), S. 141-149 
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    Notes: Abstract It was hypothesized in an earlier work that sensory perception can occur only when the perceiving system is uncertain about the nature of the event being perceived. In the absence of any uncertainty, perception will not take place. The response of the sensory afferent neuron (impulse transmission rate) was calculated using Shannon's measure of uncertainty or entropy. It will now be shown that when the event being perceived is the position and momentum of a particle, Shannon's measure of uncertainty leads to the Heisenberg Uncertainty relationship.
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    Consensusn-trees (1981)
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    Bulletin of mathematical biology 43 (1981), S. 239-244 
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    Notes: Abstract It is not unusual for several classifications to be given for the same collection of objects. We present a method, called majority rule, which can be used to define a consensus of these classifications. We also discuss some mathematical properties of this consensus tree.
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    Morphogen pattern formation and development in growth (1981)
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    Bulletin of mathematical biology 43 (1981), S. 259-270 
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    Notes: Abstract The dependence of the spatial concentration profiles of morphogens on a characteristic dimension is obtained by continuation techniques for Gierer and Meinhardt's activator-inhibitor model of morphogenesis. The study of the behaviour of the system during growth, where the linear and exponential increase of the characteristic dimension is considered, revealed that more complex patterns of morphogen spatial concentrations appear regularly in a reproducible way.
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    Simulation experiments with a model for cellular growth (1981)
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    Bulletin of mathematical biology 43 (1981), S. 271-278 
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    Notes: Abstract Computer models have been used by various authors to simulate both the growth of normal cellular tissue and the development of cancerous cells within normal tissue. As these models were the result of considerable idealization, the purpose of the present paper is to propose a model in which the degree of simplification is relaxed: the features of simultaneous growth, and cell growth whose rate depends on the free absorbing periphery of the cell are introduced. Simulation experiments have been conducted using the model, and the results are presented.
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    Complementary extremum principles for a nonlinear model of heat conduction in the human head (1981)
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    Bulletin of mathematical biology 43 (1981), S. 341-346 
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    Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for a nonlinear model of heat conduction in the human head. Accurate variational solutions are obtained in illustrative calculations. The effect of nonlinearity is seen to be significant from a comparison with the linearized model.
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    On Zeeman's equations for the nerve impulse (1981)
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    Bulletin of mathematical biology 43 (1981), S. 279-325 
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    Notes: Abstract A model for the nerve impulse due to Zeeman (1972) and based on catastrophe theory is compared with alternative models and criticisms of Zeeman's model by Sussmann and Zahler (1977, 1978) are assessed. The criticisms of Zeeman's motivation for his model are found to carry some weight. Sussmann and Zahler (1977, 1978) list numerous features of Zeeman's model which, they state, are not in agreement with experiment. These statements as they stand are largely erroneous, and the model still remains to be tested by a critical series of experiments. However, a detailed analysis reveals defects in Zeeman's model, not among those claimed by Sussmann and Zahler, showing that the explicit equations of the model cannot be correct. The possibility of a modified approach along similar lines and its ultimate adoption remains open.
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    Asymptotic analysis of the Michaelis-Menten reaction equations (1981)
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    Bulletin of mathematical biology 43 (1981), S. 375-388 
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    Notes: Abstract The irreversible Michaelis-Menten reaction is studied by the use of the method of multiple scales. Three stages of the reaction are identified, one of which is studied in detail. The results are compared with those of two earlier analyses.
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    Pulse evolution on coupled nerve fibres (1981)
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    Bulletin of mathematical biology 43 (1981), S. 389-400 
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    Notes: Abstract A numerical study of the coupled nerve fibre problem is given which verifies and extends the perturbation theory of Luzader. Pulses on adjacent fibres can couple together with two possible stable pulse separations.
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    Field-induced forces at dielectric interfaces as a possible mechanism of RF hearing effects (1981)
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    Bulletin of mathematical biology 43 (1981), S. 401-413 
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    Notes: Abstract A possible mechanism for effects of microwave radiation on the auditory system is the generation of field-induced forces at interfaces that divide materials of dissimilar electrical properties. A general expression for these “Maxwell stresses” is derived and then used to calculate the approximate magnitude of field-induced force within the organ of Corti during microwave exposure. Comparison of the results with data on the force needed to excite cochlear hair cells indicates auditory responses could be evoked by this mechanism at power densities near the threshold of rf hearing sensations.
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    The construction of homogeneous nets (1981)
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    Bulletin of mathematical biology 43 (1981), S. 415-426 
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    Notes: Abstract A definition of homogeneity for neural networks is given which permits their construction as group quotients. The significance of this for neural dynamics is discussed.
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    Contractile analysis and kinetic energy of the myocardium (1981)
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    Bulletin of mathematical biology 43 (1981), S. 447-461 
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    Notes: Abstract The left ventricle is represented as a cylinder contracting both radially and longitudinally. A simple method is indicated to derive an expression for the rate of change of the kinetic energy of this three-dimensional model, which quantity can be used as an index for the study of the contractile behaviour of the myocardium. An application to the study of muscle mechanics is also indicated.
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    Limit cycles in periodically perturbed population systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 463-485 
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    Notes: Abstract A perturbation method is proposed to calculate approximately the limit cycle type nonequilibrium steady-state resulting from periodic perturbation of coefficients of stable population systems; the two species Lotka-Volterra competition system is explicity studied and the results are formulated for general multi-species population systems. Avoidance of competitive or other types of exclusion of species in a periodic environment is indicated.
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    On a unified approach to pattern generation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 513-516 
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    Stochastic models for an enzyme reaction in an open linear system (1971)
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    Bulletin of mathematical biology 33 (1971), S. 97-115 
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    Notes: Abstract A stochastic model is developed for an enzyme reaction in an open linear system. The proposed model assumes that the open system maintains the concentration of substrate and inhibitor at constant levels and that the product molecules are removed from the system by a first order reaction. Stochastic models for several enzyme reactions occurring in this open system are shown to correspond to special cases of theGI/M/∞ queue. Takács’ (1958) results for this queueing system are used to obtain the stochastic properties of the enzyme systems. A specific model we studied assumed completely competitive inhibition in an open system. The stationary distribution for the number of product molecules in the system is obtained. The enzyme reaction which incorporated the “intermediate chain hypothesis” can also be investigated by the queueing theory approach. It is shown that for this open system, if the model which incorporated the intermediate chain hypothesis has the same deterministic properties as the Michaelis-Menten model, then the latter has greater stochastic variation than the former.
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    International Conference on Stochastic Point Processes (1971)
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    Bulletin of mathematical biology 33 (1971), S. 153-153 
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    An integral method for the analysis of blood flow (1971)
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    Bulletin of mathematical biology 33 (1971), S. 117-128 
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    Notes: Abstract The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.
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    Methode d’objectivation d’organisations temporelles privilégiées d’activités neuronales unitaires (1971)
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    Bulletin of mathematical biology 33 (1971), S. 129-151 
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    Notes: Résumé Le but de ce travail est la mise en évidence d’éventuels “patterns” temporels privilégiés de potentiels d’action neuronaux masqués par la superposition d’une activité aléatoire. Dans la première partie, on propose un modèle susceptible de rendre compte de cette activité aléatoire. Dans la seconde, on expose une méthode d’extraction des patterns privilégiés, compatible avec les paramètres du modèle neuronal proposé. Son algorithme fait notamment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, ment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, être appliquée à l’étude de séries temporelles d’événements dans des domaines très divers.
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    International Symposium on Mathematical Biology (1971)
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    Bulletin of mathematical biology 33 (1971), S. 155-156 
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    Molecular set theory. III: The wide sense kinetics of molecular sets (1971)
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    Bulletin of mathematical biology 33 (1971), S. 355-372 
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    Notes: Abstract An effort is made to begin widening the scope of kinetics by merging the concepts and point of view of molecular set theory with the stochastic approach to kinetics, beginning with the simplest unimolecular molecular set transformation. In this spirit the new concept ofmolecular set variable is introduced as the basic unit of kinetics as opposed to simply the traditionalconcentration (or cardinality) unit, connoting that the composition as well as the size of a molecular set are significant dynamic features of a system. The changes in state (or “value”) of the molecular set variable are characterized by a Markovian stochastic process and the relationship between this process and the corresponding unimolecular process for the concentration variable introduced earlier is discussed. The possible role of molecular set theory in terms of the underlying biomathematical structure of relational biology is also considered.
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    Contribution a l’etude de la forme des plantes: discussion d’un modele de ramification (1971)
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    Bulletin of mathematical biology 33 (1971), S. 387-401 
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    Notes: Abstract The problem of the forms of plants and models of branching are discussed using experimental data on the mistletoe. The number of branches by division, the distribution of divisions with regard to the number of branches per division and to the level of division, the geometrical characters of branches according to the level of division and the host, the stability of model are studied. One gives an interpretation of the model of branching as a model of growth.
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    A quantum model for controlled enzymic reactions (1971)
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    Bulletin of mathematical biology 33 (1971), S. 373-386 
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    Notes: Abstract A quantum model for the general enzymic reaction,E+S ⇌ ES → P, is presented, starting with the assumptions that any chemical substanceS, which may be a substrate for a particularE (S)-enzyme is a microphysical system and any enzymeE-molecule, capable of interacting with anS-substrate is a “measuring system” which will “measure” one or more of theS-observables. According to the above assumptions a stochastic model of the reaction is constructed and a computer simulation of the steady state performed. The results thus obtained predicted fluctuations in the enzymic reaction rate, function of the substrate “perturbation”. On an experimental basis it is demonstrated that the irradiation of an enzymic substrate with low energies results in the inducement of a dose-dependent oscillatory behavior in the corresponding enzymic reaction rate. In the reaction type, the oscillations thus induced in theE-activity by the corresponding substrates are out-of-phase, realizing a biochemical discriminating net. Likewise, in an $$S_1 \xleftarrow{{E_1 }}S\xrightarrow{{E_2 }}S_2$$ reaction type, the oscillations induced by the irradiatedS-substrate in the activities of the respective enzyme, realize a biochemical switching net.
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    The myocardium as a composite material: Analysis (1972)
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    Bulletin of mathematical biology 34 (1972), S. 173-204 
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    Notes: Abstract As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.
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    A note on epilepsies and similar disorders (1972)
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    Bulletin of mathematical biology 34 (1972), S. 213-222 
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    Notes: Abstract This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.
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    Suggestions for a mathematical model of a pathological neuron with spontaneous repetitive discharges (1972)
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    Bulletin of mathematical biology 34 (1972), S. 223-230 
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    Notes: Abstract A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.
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    Structure and kinetic behavior of linear multicompartment systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 277-291 
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    Notes: Abstract The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system.
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    A dynamical approach to chemical kinetics: Mass-action laws as generalized Riccati equations (1972)
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    Bulletin of mathematical biology 34 (1972), S. 243-275 
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    Notes: Abstract It is shown how the fundamental laws of chemical kinetics for either open or closed systems with an arbitrarily large number of reactants can be represented as a system of Riccati-like differential equations. Through the use of a concise tensor notation, it is shown when and how the differential system is exactly reducible to linear form, a reduction without approximation that parallels the well-known similar reduction of a single simle Riccati equation. An example is worked out to show how open kinetics can lead to oscillatory chemical concentrations of the Change-Higgins type. The biologically central problem of great chemical speciation is discussed from the viewpoint of Gibbs ensemble theory within the linearized kinetics and, approximately, within the starting nonlinear kinetics where it is shown roughly how to estimate, from an overall temperature-like parameter characterizing the whole system, mean chemical levels and mean frequencies of oscillation, and where a gross oscillation of the total mass is estimated in terms of an anharmonic oscillator whose general structure is fixed from the structure of the chemical kinetic laws.
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    Note on oscillations in a closed chain of compartments with linear, unidirectional transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 293-296 
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    Notes: Abstract A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.
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    Pourquoi deux sexes? (1972)
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    Bulletin of mathematical biology 34 (1972), S. 297-304 
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    Notes: Abstract This paper is a concrete approach to the problem of the number of the sexes. We try to imagine—on the example of three sexes—the mechanisms which would have to accompany a reproduction with several sexes. We have limited our study to the monohybridism, dihybridism and determinism of the sex.
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    Cilia transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 325-335 
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    Notes: Abstract An analysis of the effect of cilia on fluid transport in tubules is presented. The applicability of the results for the flow rates observed in the ductus efferentes of the male tract is discussed.
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    Quantitative comparison of dipole models for steady-state currents in excitable membranes (1972)
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    Bulletin of mathematical biology 34 (1972), S. 305-324 
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    Notes: Abstract The dipole models for steady-state currents in excitable membranes of Arndt, Bond and Roper and of Hamel and Zimmerman are compared by fitting the equations to the data of Gilbert and Ehrenstein. The more complex Hammel and Zimmerman model does not fit the data as well as does the simpler Arndt, Bond and Reper model. When fitting the data, the Hammel and Zimmerman current equation reduces to the Arndt, Bond and Roper current equation because of the values assumed by the parameters. An interpretation is given for the parameter values obtained with the Arndt, Bond and Roper model.
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    On the decomposition of a dynamical system into non-interacting subsystems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 337-341 
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    Notes: Abstract It is shown that, under rather general conditions, it is possible to formally decompose the dynamics of ann-dimensional dynamical system into a number of non-interacting subsystems. It is shown that these decompositions are in general not simply related to the kinds of observational procedures in terms of which the original state variables of the system are defined. Some consequences of this construction for reductionism in biology are discussed.
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    Self-reproduction and serial message transfer: Two related problems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 343-353 
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    Notes: Abstract For a certain class of physical machines, termed “structure-determined,” the problem of self-reproduction can be reduced to the problem of serial message reproduction. Serial message reproduction however presupposes a sort of “open system” constraint. This leads to the principle of pseudo, or exogenously standardized, respectively, self-reproduction. It seems to be consistent with both chemical and biological self-reproduction. It thus may reflect a general principle of biological design. The proposed principle is a physico chemical analog to Robert Rosen's abstract relational self-reproduction constraint.
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    Active muscle fiber as an equivalent cardiac current generator (1972)
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    Bulletin of mathematical biology 34 (1972), S. 355-377 
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    Notes: Abstract Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.
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    Analytical solutions for transient thermal behavior in biological systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 413-418 
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    Notes: Abstract Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.
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    Simultaneous diffusion and convection in single breath lung washout (1972)
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    Bulletin of mathematical biology 34 (1972), S. 393-412 
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    Notes: Abstract Two mathematical models of pulmonary single breath gas washout (one analytic, one numerical) are developed and their predictions compared with experimental data on human subjects. Weibel's 23 generation symmetric anatomical model is used as a guide to bronchial tree geometry. Experimental plots of nitrogen concentration versus volume expired, dead space versus breath holding time, and dead space versus tidal volume are compared with plots predicted by the models. Agreement is good. A plot of nitrogen concentration in the airways as predicted by the numerical model at different times during inhalation and exhalation of a single breath of oxygen is shown. Model predictions for changes in dead space with changes in washout gas and expiratory flow rate are discussed. Use of the analytic model for obtaining average values of the path length from mouth to alveoli in a given subject is discussed. To the extent of their agreement with experiment, the models provide a sound physical basis for the correlation of airway structure and function.
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    Announcement (1972)
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    Bulletin of mathematical biology 34 (1972), S. 429-429 
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    Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces (1972)
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    Bulletin of mathematical biology 34 (1972), S. 419-427 
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    Notes: Abstract The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m 1 exp (n 1x) −m 2 exp (n 2 x), wherem 1,m 2,n 1 andn 2 are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.
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    Limits on the computing power of biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 59-67 
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    Notes: Abstract The theory of computational complexity and certain explicitly-stated hypotheses imply limitations on the information processing power of biological systems. Parallelism, special purpose organization, and analog mechanisms may provide speedup critical for life processes, but have little power in the face of exponential growth. We show that “polynomially simulatable” biological systems cannot exhibit dynamic behavior which produces the solution of an intractable problem. The argument implies that parallelism does not allow biological systems to defeat the exponential explosion, but rather is important because it allows polynomial time algorithms to be used more efficiently.
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    A topological study of the base sequences of MS2 and ϕX174 genoma (1981)
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    Bulletin of mathematical biology 43 (1981), S. 81-88 
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    Notes: Abstract A correlation matrix analysis is applied to the base sequence of MS2 and ϕX174 in comparison with sets of simulated sequences with different degrees of constaint Significant differences between a codified sequence, and a statistical one in terms of the “correlation matrix” for sets of different length cannot be found. This result is analysed in terms of nucleotide sequences with different levels of informational content.
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    Volume of the conducting airways calculated from morphometric parameters (1981)
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    Bulletin of mathematical biology 43 (1981), S. 101-109 
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    Notes: Abstract A method of calculating the volume of a tree distal to a cut at the origin of a branch, using branching, diameter and length ratios, has been developed. The method was applied to bronchial tree casts from human, dog, sheep, hamster, and rat lungs. It was found that the exponenta in the equation weight=k×diameter a is approximately equal to 3.0 in sheep lung casts, as found by Hooper (1977), but it is greater than 3.0 in casts from the other four species.
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    Variability in a single compartment system: A note on S. R. Bernard's model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 111-116 
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    Notes: Abstract In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.
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    The diffusion transit time; A simple derivation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 89-99 
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    Notes: Abstract The mean first passage time for free diffusion can be derived directly by solving a simple analogue steady state problem. In this problem the diffusion starting region is considered as a time independent source of diffusing particles and the diffusion target assumes the behaviour of a perfectly absorbing sink. It is shown here that the transit time between the source and the sink, which in this particular problem is equal to the ratio between the holdup of the system and the total flux, is identical to the Brownian movement concept of the mean first passage time for free diffusion. This established identity considerably facilitates the derivation and investigation of the timing of diffusion in complicated structures such as those commonly found in living organisms.
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    Nonlinear oscillations in biology (1981)
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    Bulletin of mathematical biology 43 (1981), S. 121-123 
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    A comment on Lin's paper on nonlinear diffusion in biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 117-120 
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    On the stochastic theory of compartments: The leaving process of the two-compartment systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 201-211 
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    Notes: Abstract In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.
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    Models on prebiotic polymer competition: A deterministic approach (1981)
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    Bulletin of mathematical biology 43 (1981), S. 213-232 
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    Notes: Abstract Two simple models are proposed and analysed, in which it is shown that the formation of a new polymer, resulting from an “error” in the template action mechanism of production of an old polymer, may compromise the stability of the initial system under specific conditions, in the context of prebiotic evolution. Autocatalysis is shown to be a “selective advantage”, enabling the “mutant” to dominate in concentration and even replace the initial polymer. The addition of a third molecule playing the role of a catalyst causes hysteresis effects.
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    Extinction probabilities in microbial predation: A birth and death approach (1981)
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    Bulletin of mathematical biology 43 (1981), S. 165-181 
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    Notes: Abstract The problem of extinction of the prey population in a microbial predator-prey interaction in a chemostat has been examined. Usual deterministic lumped parameter models were used for the dynamics of the chemostat for large numbers of the two populations; the generalized birth and death stochastic process was employed for the description of the random variations at small prey numbers. Extinction probabilities of the prey population were calculated for different holding times and chemostat volumes, and their dependence upon the growth parameters of the two populations was studied. It was found that extinction was possible when the Monod model was used for the specific growth rate of the predators as a function of the prey number density. On the other hand, the decrease of the feeding activity of the predators at low prey densities predicted by the multiple saturation model acts as a regulatory factor that prevents extinction of the prey. In view of the fact that extinction of the prey has never been observed in the laboratory, the latter model seems more appropriate to describe the dynamics of microbial predation.
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    Prediction of the effect of the choroidal circulation on the retinal temperature during illumination (1981)
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    Bulletin of mathematical biology 43 (1981), S. 233-238 
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    Notes: Abstract During exposures of the eye to light, the choroidal circulation may have a regulatory influence on the retinal temperature. This is investigated using a mathematical model and a finite-difference technique. It is predicted that the choroidal blood flow a small effect on retinal temperature, which may be important.
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    Spatially localized interactive neural populations—I. A mathematical model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 427-446 
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    Notes: Abstract A probabilistic model of a spatially localized, mutually exitatory (inhibitory) population of neurons is formulated to help explain average evoked potential and post-stimulus time histogram measurements. The model is based on the stochastic activity of single neurons within interactive masses of neurons which exhibit co-operative behavior. Macrostate variables corresponding to the above measurements are related through the model to features of neural operation at the individual and ensemble level. Steady-state solution are obtained and their physiological implications are discussed.
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    A one-compartment model with stochastic parameter (1981)
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    Bulletin of mathematical biology 43 (1981), S. 503-512 
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    Notes: Abstract We consider a one-compartment system with stochastic transfer rate characterized either by Gaussian or by two-level jump process and study the time evolution of the (statistical) moments of the (random) amount of the substance present in the system. The effect of the coloured as well as of the white noise is investigated and it is found that the presence of stochasticity in the transfer rate parameter increases the relaxation time of the system. Finally, we obtain the conditions for the stability of the system in the moment sense.
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    A dose-response model for estimating lifetime tumor risks when cell killing occurs (1981)
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    Bulletin of mathematical biology 43 (1981), S. 487-501 
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    Notes: Abstract A model is described in which damage to a single intracellular locus can lead to a tumorigenic transformation. Assuming a large number of independent intracellular loci to be at risk and assuming that damage to a locus sufficient to cause a tumorigenic transformation occurs with probability greater than zero for all doses greater than zero, leads to the use of the Weibull distribution to characterize the probability of a nonspontaneous tumorigenic cellular transformation occurring after exposure to a given dose of carcinogen. The excess lifetime tumor incidence (i.e., the proportion of tumor bearers) above the spontaneous incidence is used as an estimate of the non-spontaneous incidence and is characterized by a tumor incidence function that represents the probability of occurrence of one or more non-spontaneous tumorigenic cellular transformations amongN(D) independent surviving cells per individual, after exposure to a doseD of carcinogen. The tumor incidence function is fitted to published data for the excess tumor incidence after exposure of animals or humans to ionizing radiation and after exposure of animals to chemical carcinogens.
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    A stochastic model for then-compartment irreversible system (1981)
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    Bulletin of mathematical biology 43 (1981), S. 549-561 
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    Notes: Abstract This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.
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    Pulsatile blood flow in a rigid pulmonary alveolar sheet with porous walls (1981)
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    Bulletin of mathematical biology 43 (1981), S. 563-577 
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    Notes: Abstract This paper deals with the pulsatile blood flow in the lung alveolar sheets by idealizing each of them as a channel covered by porous media. As the blood flow in the lung is of low Reynolds number, a creeping flow is assumed in the channel. The analytical and numerical results for the velocity and pressure distribution in the porous medium are presented. The effect of an imposed slip condition is also studied. Comparisons with the corresponding results for the steady-state case are made at the end.
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    Dynamic structure theory: A structural approach to social and biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 579-591 
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    Notes: Abstract The relationships that define the structure of a given ecosystem, social system, or even a physiological function can only exist if certain parameters are confined to a certain range of values. As the values change and exceed this given range the relationships are forced to change, and so produce a new pattern of relationships. The concept of a dynamic structure captures this potential for structural change in relation to a set of parameters. The precise definition of structure and allowable transformation constitutes the definition of a category. The total range of parameters associated with all the relevant structures provides a parameter space which is assumed to be a manifold. Maps with extra structure from the manifold to the category define dynamic structures. The domain of differential dynamic systems is the manifold, and a flow or movement across the manifold is associated with a series of structural transformations in the category. In some cases a structure outruns its parameter range, to be faced with an obstruction—an absence of possible transformations. Ways of studying such “obstructions” are considered along with the related problem of extending a dynamic structure beyond a previously given set of parameters. The cost or resistance of transformations is also studied. The concepts of dynamic structures are illustrated by the structural change of food webs and they are used in a necessarily qualitative fashion to study dominance structures of social orders and finally to speculate on the qualitative nature of evolutionary change of functional aspects of organisms.
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    Peak drug levels in linear pharmacokinetic systems—II. Conditions for a paradoxical injection rate effect with rectangular input (1981)
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    Bulletin of mathematical biology 43 (1981), S. 705-715 
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    Notes: Abstract The preceding paper (Thorn, 1981) has shown that in a linear pharmacokinetic system with a multimodal impulse response the peak drug level may sometimes be smaller with slower rates of injection. This paper presents two theorems on this paradoxical injection rate effect where the injection is a constant infusion of finite duration. The first theorem establishes a graphical method for determining whether a given impulse response will give a paradoxical injection rate effect; and the second establishes that the maximum paradoxical increase in peak drug level is by a factor of two. It is further shown that in order to approach this maximum paradoxical increase the impulse response must contain two isolated, sharp, narrow pulses of approximately equal area. Some examples of bimodal arterial dye-dilution curves from the literature are discussed as impulse responses; and there is also a discussion of the behavior of drug level maxima and minima at different injection rates.
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    Peak drug levels in linear pharmacokinetic systems—I. Effect of rate of injection (1981)
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    Bulletin of mathematical biology 43 (1981), S. 693-703 
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    Notes: Abstract This paper presents three theorems on the peak drug levels that result from injection into a linear pharmacokinetic system. As a preliminary, the “rate of injection” is defined in terms of time expansion or time contraction of the injection function (input). The first theorem then states that the peak drug level will not be greater when the rate of injection is slow than when it is fast, if the impulse response is unimodal. The second theorem sets limits for the time of the maximum drug level, in relation to the time of the maximum of the (unimodal) impulse response and the duration of the input. The third theorem defines conditions which assure a definitely lower peak drug level if the rate of injection is slower. A graphical method is suggested for determining the times and magnitudes of the peak drug levels that result from constant infusions of a fixed dose at different rates. An example is provided to show that if the impulse response is multimodal then the peak drug level may sometimes increase with a decrease in the injection rate.
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    Basic studies on metabolic steady state. Incompletely mixed systems (1971)
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    Bulletin of mathematical biology 33 (1971), S. 403-412 
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    Notes: Abstract This paper is the second of a pair dealing with some mathematical properties of metabolic steady state. An investigator wishing to compute the rate of appearance and/or disappearance of a metabolite in steady state within an intact biological system will usually appeal to a method involving radioactive tracers. It is shown that while the investigator’s choice of the mode of tracer administration (constant infusion or single injection) is largely arbitrary, the mathematical interpretation of the results may depend upon the presence or absence of gradients in certain of the variables of the system. The latter will be the case if the system is sampled at a point within the distribution space of the metabolite which is not a source point but is otherwise arbitrary. In order to deduce a formula which gives the required rate, he must have knowledge of the gradient of concentration of the traced substance, and sometimes of the gradient of specific activity.
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    Some mathematical aspects of the medical diagnostic process. I: a general mathematical model (1971)
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    Bulletin of mathematical biology 33 (1971), S. 413-424 
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    Notes: Abstract Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.
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    Forecasting and chronic illness (1971)
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    Bulletin of mathematical biology 33 (1971), S. 425-437 
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    Notes: Abstract Biomedical data in the form of series of observations made on a single process at regular intervals constitute a discrete time series and are eligible for time series methods of analysis. The models yielded by this analysis provide the framework within which exponential smoothing methods may operate on the data to provide recurrent forecasts of future states of the process. Because the forecasts may be made on an individual basis and are sensitive to the past behavior of the individual process, the methods are presented as being potentially of great utility in the management of chronic and progressive illnesses. When incorporated into automated testing and diagnostic systems, the forecasting method will provide the capability of making prognoses for large numbers of individuals, quickly, routinely and reproducibly.
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    The determination of cell-cycle parameters from measurements of the fraction of labeled mitoses (1972)
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    Bulletin of mathematical biology 34 (1972), S. 1-12 
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    Notes: Abstract A method is described for estimating cell-cycle parameters from experimental fraction-of-labeled-mitoses measurements. The method is closely related to that of J. C. Barrett (1970) but is based on the analysis of Brockwell and Trucco (1970) which takes into account population growth in the calculation of theoreticalFLM-functions. Several sets of experimental data are analyzed, among them the data for the Marshall tumor considered by Barrett. It is found that population growth has a small but nevertheless detectable effect on the estimates of the cell parameters.
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    URL: http://dx.doi.org/10.1007/BF02477020
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    Transfer times across the human body (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 33-44 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The radioactivity disappearance curves of glucose-6-14C albumin-I131 after a single injection of tracer into a human subject have been determined in detail, particularly at early time intervals. The curves, expressed as sums of exponentials, have been analyzed as the infinite sum of convolutions of single passage time densities. The resultant transfer time distribution of a single circulatory pass allows examination of all delays in the system no matter how long they take. The structural detail evident by this means and the long mean time of a single pass of glucose (〉5 min) supports the thesis that factors other than rapid and uniform diffusion play a role in the extravascular movements of glucose molecules.
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    URL: http://dx.doi.org/10.1007/BF02477022
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    A bisexual multitype branching process with applications in population genetics (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 13-31 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A bisexual multiple branching process is studied. Consider a population with respect to three genotypes in both the female and male populations and let $$X(n) = \left\langle {X_1 (n), X_2 (n), X_3 (n)} \right\rangle and Y(n) = \left\langle {Y_1 (n), Y_2 (n), Y_3 (n)} \right\rangle$$ be random vectors giving the number of females and males (respectively) of each genotype in generationn. The mating of females and males is accommodated in the model withZ ij (n) representing the number of females of theith genotype mated with a male of thejth genotype in generationn. The mating system is such that a female may be mated to only one male but a male may be mated with more than one female. By arranging the nine random variablesZ ij (n),i, j=1, 2, 3, in a 1×9, vectorZ(n) it is shown that under certain conditions there is a positive constant ϱ such that when ϱ〉1 the vectorsZ n /ρn,X n /ρn andY n /ρn converge almost surely asn→∞ to random vectors with fixed directions. The paper is divided into four sections. In section 1 the model is described in detail and its potential applications to population genetics are discussed. In section 2, the generating function of the transition probabilities of theZ-process are derived. Section3 is devoted to the study of the limiting behavior of the first and second moments of theZ-process, and in section4 the results of section3 are utilized to study the behavior of the random vectorsZ(n),X(n) andY(n) asn→∞.
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    URL: http://dx.doi.org/10.1007/BF02477021
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    Theory of initial current density in membrane voltage-clamp experiments (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 45-52 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Herein we show that the voltage-clamp current density at zero time calculated from electrodiffusion equations is linear in the clamping voltage for a simple membrane (no charge structure) and for a membrae with fixed charges. Such membranes are nonexcitable. Excitable membranes can be represented by a homogeneous membrane with dipole layers at the surface. In this case the initial current density will be linear in the clamping voltage if a critical field for a dipole layer reorientation is not passed through in changing from holding to clamping potential. Otherwise, deviation from nonlinearity may occur. This is in agreement with experimental data for the squid giant axon.
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    URL: http://dx.doi.org/10.1007/BF02477023
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    A note on sporadic nonperiodic oscillations in the activity of the endocrine system and on the endocrine—Nervous system feedback (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 65-69 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract By generalizing a previous paper on periodicities in the endocrine system (Bull. Math. Biophysics,30, 735–749, 1968), it is shown that nonperiodic, sporadic oscillations in the system are also possible. A procedure of describing the feedback mechanism between the endocrine system and the central nervous system is suggested. It is shown that the combined system: endocrine—CNS, also may show sporadic fluctuations.
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    URL: http://dx.doi.org/10.1007/BF02477025
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    Wall shear in pulsatile flow (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 79-86 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.
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    URL: http://dx.doi.org/10.1007/BF02477027
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    A neural circuit which exhibits some features of epileptic attacks (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 71-78 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A neural model based on a generalization of a model proposed in 1938 in the first edition of the author'sMathematical Biophysics (Chicago: Univ. of Chicago Press) is described. It possesses the property that due to some endogenous or exogenous stimulus, which may be of random nature, a pathway may suddenly begin to fire spontaneously. This spontaneous firing may either gradually spread over other pathways and eventually cease, or it may remain localized within one or a few pathways and then cease. Which of the two types of events occurs depends on the values of a number of parameters. The case of spreading reminds one of Jacksonian epilepsy.
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    URL: http://dx.doi.org/10.1007/BF02477026
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    Diffusion exchange between a membrane-bounded sphere and its surrounding (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 93-102 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.
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    URL: http://dx.doi.org/10.1007/BF02477029
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    Determination of reaction rate constants in the mammillary system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 87-92 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.
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    Some biochemical threshold mechanisms (1972)
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    Bulletin of mathematical biology 34 (1972), S. 103-112 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Certain arrangements of enzymatic (bimolecular) subsystems lead to characteristic threshold-type response. Two simple cases of such systems are studied here in terms of steady state behavior and explicit relationships between system and curve parameters. It is found that the curvature of the threshold curve is directly related to the equivalent Michaelis constant and, in the case of saturated threshold curve, the slope of the curve at the idealized threshold is limited by the ratio of saturation to threshold. This slope may be appreciably increased up to a stepwise response at the threshold if a multisubstrate complex of the enzyme is the only species which affects the enzyme mediated transport.
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    URL: http://dx.doi.org/10.1007/BF02477030
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    Theory of neural mediation of intraocular dynamics (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 113-148 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Many experimental studies have indicated that the intraocular pressure is subject to mediation by adrenergic mechanisms affecting both the rate of formation of the aqueous humor and the resistance of the pathway through which the aqueous humor flows out of the eye. Thus, for example, the role of adrenergic drugs in glaucoma therapy is well known. How the mediation is accomplished has not been clarified in detail. Several possible mechanisms have been suggested, and all may indeed be involved. The present study is concerned with the basis and mathematical formulation of one of them and the consequences with respect to aqueous dynamics. The analysis leads to expressions for the aqueous outflow resistance and the formation rate, as well as other quantities of interest. The theoretical behavior is shown to compare favorably with the results of infusion studies and various other experiments, and to provide a unified picture of much of the pressure-flow behavior of both the living and the dead eye.
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    URL: http://dx.doi.org/10.1007/BF02477031
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    Eighth International Congress of Electroencephalography and Clinical Neurophysiology (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 149-150 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477032
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